Division Tricks
Divisibility by 9 – The Digit Sum Again
The same digit-adding idea as for 3, but the total must be a multiple of 9.
This page explains the rule, shows why it works, and walks through worked examples. Then you can try it yourself and check your understanding with practice questions. Use the previous and next links at the bottom to move through the tricks in order.
The rule
The Rule: If the sum of all the digits of a number is divisible by 9, then the number itself is divisible by 9.
Why it works
Why It Works: This works for exactly the same reason the rule for 3 does — just with 9 instead. Every place value (1, 10, 100 …) is also one more than a multiple of 9, so once again, adding up the digits collects all those tiny leftovers, and the digit sum always carries the same remainder as the full number.
Formal Proof (for advanced learners)
Since 10 ≡ 1 (mod 9), it follows that 10^i ≡ 1 (mod 9) for every power i. So N = ∑ a_i·10^i ≡ ∑ a_i (mod 9), meaning N and its digit sum always share the same remainder when divided by 9, and so are either both divisible by 9 or neither is.
Examples
- 729 → 7 + 2 + 9 = 18, divisible by 9.
- 981 → 9 + 8 + 1 = 18, divisible by 9.
- 514 → 5 + 1 + 4 = 10, not divisible by 9.
Step by step
Example A: is 7893 divisible by 9?
- Add the digits: 7 + 8 + 9 + 3 = 27.
- 27 is divisible by 9.
- So 7893 is divisible by 9.
Example B: is 4562 divisible by 9?
- Add the digits: 4 + 5 + 6 + 2 = 17.
- 17 is not divisible by 9.
- So 4562 is not divisible by 9.
Try it yourself
Now it is your turn. A new random question appears each time. Choose True or False. If you are right, we show how the trick got you there, step by step. If you are not, you will see the full working two ways, the shortcut and the usual method, so you can spot exactly where your answer differed. Press New Question as often as you like: a few quick rounds is the best way to make the trick feel automatic.
Practice questions
Have a go on paper first, then tap Show answer to check your method.
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Is 81 divisible by 9?
Show answer
Yes. Add the digits: 8 + 1 = 9. 9 is divisible by 9. So 81 is divisible by 9.
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Is 234 divisible by 9?
Show answer
Yes. Add the digits: 2 + 3 + 4 = 9. 9 is divisible by 9. So 234 is divisible by 9.
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Is 475 divisible by 9?
Show answer
No. Add the digits: 4 + 7 + 5 = 16. 16 is not divisible by 9. So 475 is not divisible by 9.
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Is 6318 divisible by 9?
Show answer
Yes. Add the digits: 6 + 3 + 1 + 8 = 18. 18 is divisible by 9. So 6318 is divisible by 9.
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Can 12345 cookies be shared equally among 9 children with none left over?
Show answer
No. Add the digits: 1 + 2 + 3 + 4 + 5 = 15. 15 is not divisible by 9. So 12345 is not divisible by 9.
Tips and watch-outs
- Keep adding digits until you get one digit: for a multiple of 9 that digit is always 9. Try 6318 → 18 → 9.
- Every multiple of 9 is also a multiple of 3, but not the other way round: 12 has digit sum 3 (divisible by 3) but not 9.